Is there any quality difference between the two? Dry malt extract is way more convenient for me, but are there any downsides? Can I use just DME (and no liquid malt extract) with no problems in some recipe?

In my experience there's no effect on the quality of the beer, and I've also found DME to be the preferred ingredient if I have a choice, because of the convenience. You'll also probably find a bigger variety of DME (sometimes even specific to a single grain) at your brew shop because it's the more popular of the two, so developing your own recipes is very easy.

Other than the extra water in LME there's no quality difference in my experience. There is a very slight difference in handling.

LME is hard to use a partial quantity as it sticks to everything, making it difficult to measure out accurate weight or volume. For that reason I prefer DME, though pouring DME into a steaming wort does cause some of the crystals in the packaging to absorb the steam and turn sticky (but it's negligible)